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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数练习3.3 - 柯西积分公式 }
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\date{2024 年 4 月 22 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设区域 $D$ 的边界是周线 $C$, 设函数 $f(z)$ 在 $D$ 内解析，在 $\overline{D}=D\cup C$ 上连续。设 $z\in D$. 
证明柯西积分公式 $$f(z) = \frac{1}{2\pi i} \int_C\frac{f(\zeta)}{\zeta-z}d\zeta. $$

\vspace{0.2cm}

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\item  %Problem 02
设 $C$ 为圆周 $|\xi|=2$. 使用柯西积分公式，计算 $$\int_C \frac{\zeta d\zeta}{(9-\zeta^2)(\zeta+i)}.$$

\vspace{0.2cm}

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\item  %Problem 03
设 $C$ 为圆周 $|\xi-i|=\frac{1}{2}$. 使用柯西积分公式，计算 $$\int_C \frac{dz}{z(1+z^2)}.$$

\vspace{0.2cm}

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\item  %Problem 04
设函数 $f(z)$ 在圆 $|z-z_0|<R$ 内解析，在闭圆 $|z-z_0|\le R$ 内连续。证明%平均值定理  
$$f(z_0) = \frac{1}{2\pi}\int_{0}^{2\pi} f(z_0+Re^{it})dt. $$

\vspace{0.2cm}

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\item  %Problem 05
设函数 $f(z)$ 在闭圆 $|z|\le R$ 上解析，如果存在 $a>0$, 使得 $|f(0)|<a$, 且当 $|z|=R$ 时，有 $|f(z)|>a$. 
证明存在 $|z_0|<R$ 使得 $f(z_0)=0$. 
\vspace{0.2cm}

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\item  %Problem 06
设区域 $D$ 的边界是周线 $C$, 设函数 $f(z)$ 在 $D$ 内解析，在 $\overline{D}=D\cup C$ 上连续。设 $z\in D$. 
使用柯西积分公式，证明 $$f'(z) = \frac{1}{2\pi i} \int_C\frac{f(\zeta)}{(\zeta-z)^2}d\zeta. $$

\vspace{0.2cm}

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\item  %Problem 07
设 $C$ 为圆周 $|z-i|=1$, 使用解析函数求导的柯西积分公式，计算积分 $$\int_C \frac{\cos(z)dz}{(z-i)^3}. $$

\vspace{0.2cm}

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\item  %Problem 08
设 $C$ 为圆周 $|z|=r>1$, 计算积分 $$\int_C \frac{\exp(z)dz}{(z^2+1)^2}. $$

\vspace{0.2cm}

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\item  %Problem 09
设函数 $f(z)$ 在区域 $D$ 内解析，设圆盘 $S=\{z: |z-a|\le R\}$ 整个都落在区域 $D$ 内，记 $M$ 为 $|f(z)|$ 在圆盘 $S$ 的边界 %$\gamma=\partial S$ 
上的最大值。证明 $$|f^{(n)}(a)| \le \frac{n!}{R^n}M. $$

\vspace{0.2cm}

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\item  %Problem 10
设 $f(z)$ 是整函数，即在整个复平面上都解析。设 $f(z)$ 有界，则 $f(z)$ 是常数。

\vspace{0.2cm}

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\item  %Problem 11
设 $n\ge 1$, 设 $a_0,a_1,\cdots,a_n$ 都是复数，且 $a_0\neq 0$. 证明存在复数 $z$ 使得  
$$a_0z^n+a_1z^{n-1}+\cdots+a_{n-1}z+a_n=0. $$

\vspace{0.2cm}

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\item  %Problem 12
设函数 $f(z)$ 在单连通区域 $D$ 内连续，设对 $D$ 内任意周线 $C$ 都有 $$\int_C f(z)dz=0,$$
证明 $f(z)$ 在 $D$ 内解析。

\vspace{0.2cm}

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\item  %Problem 13
设 $C$ 为圆周 $|z|=2$, 计算积分 
$$
(1) \int_C \frac{2z^2-z+1}{z-1}dz, \hspace{1cm}  
(2) \int_C \frac{2z^2-z+1}{(z-1)^2}dz. 
$$

\vspace{0.2cm}


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\item  %Problem 14
设 $C$ 为圆周 $|z|=1$, 计算积分 
$$
(1) \int_C \frac{\exp{z}}{z}dz, \hspace{1cm}  
(2) \int_0^{\pi} \exp(\cos t)\cos(\sin t)dt. 
$$

\vspace{0.2cm}

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\item  %Problem 15
设 $C$ 为圆周 $|z|=3$, 设 $$f(z) = \int_C \frac{3\zeta^2+7\zeta+1}{\zeta -z}d\zeta. $$ 
求 $f'(1+i)$. 

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\end{enumerate}


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\end{document}

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